I am thinking about how particles attract each other, but also repel. All matter attracts via gravity, but also attract and repel each other due to the electromagnetic force, also governed via the inverse-square law, and then there's the strong and weak force.

I understand the idea that two hydrogen atoms repel, but when pushed together hard enough will attract and form helium and release energy. So this "upwards slope" of repulsion which is eventually "snaps" into attraction, like trying to roll a ball up a smooth hill with a hole in the center of it, is explained, likely, by varying forces governed by various laws. Can you help me understand it, on the level of particles?

Where I am stuck is thinking about two particles, each being attracted and repelled by two different forces each governed by the inverse-square law. This doesn't explain the "snapping" together of atoms. Do I need different math in my forces, or is it ultimately explained by more than just two particles?

5 Answers
5

First, you may become disappointed but the trully fundamental laws, as we know them today, are not written in terms of force laws. Even though the concept of force is still present
in Physics, it is not used in the way it was before and which seems to be the way you are thinking about them.

Force is nowadays synonymous of interaction and one does not seek for force laws to be used in the equation
\begin{equation}\vec{F}=\frac{d\vec{p}}{dt},\end{equation}
from where one would, ultimately obtain $\vec{r}(t)$.

The above equation summarizes classical mechanics (CM) in its Newtonian "version" (or formulation). Even classical mechanics can be done without explicitly writing a vector equation as this one.

It was the analytic formulation(s) of CM that people took as the framework for doing advances in mechanics. They are all equivalent when it comes to the classical scenario and one uses one or another formulation for several reasons. Nevertheless, in the analytic formulations of CM, instead of using forces, as the quantities encoding the interaction, one uses potentials and the equations of motion are no longer obtained from Newton's second law (at least not explicitly as in the equation above) but from a more powerful principle, which is Hamilton's principle.

Now, even though in CM one may use any formulation according to one's needs, when it comes to relativistic classical mechanics and (relativistic) quantum mechanics, it is no longer a matter of choice. There are several reasons for why this is so. A very simple one, is that will won't be able to find a force four-vector to plug in the relativistic equivalent of Newton's second law (as it is written above) other than the Lorentz force. Also, in quantum mechanics (QM), Newton's second law holds only as an average (or expectation value).

This is why, even though one still speaks about forces, it is not in the same sense as before and we don't have other kinds of inverse-distance laws (or any other kind of vetor force laws) for the other fundamental interactions. Even the so-called potentials are not quite the same animals as in CM.

About What laws govern the fundamental forces of nature?, have a look at here.

Even the problems we try to solve with more fundamental physics are not quite the same as in CM. It is more about cross sections and decay rates than about describing the motion of individual particles (though that can be done to some extent).

I believe the particular phenomenon you are interested in is nuclear fusion. It is ultimately described in terms of electromagnetic and strong interactions and, even though in practice people may describe it in terms of more effective nuclear forces, it is still all done in the framework of relativistic quantum mechanics / quantum field theory and you won't find force laws.

To summarize: there are no force laws aside the ones of classical physics (Newton's gravitation law, Coulomb's electrostatic force and Lorentz force and some others).

This is great. Thank you. I will be giving the answer to Brandon Enright above but your answer has also helped a lot. EDIT: I will give the answer to you since you answered it first.
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XonatronNov 22 '13 at 18:15

Your figure represents not two inverse-square potentials but something like the Lennard-Jones potential $V(r)= Cr^{-12}-C'r^{-6}$. The latter is a model for
the van der Waals potential between two neutral spherical particles when they are at distance $r$.

The force is the derivative. hence at close distance, they repel each other, further apart they attract each other (and very far apart they don't notice each other significantly). Thus if close enough that the approximation is valid, their distance will oscillate between an attracting and a repelling distance, until dissipation will bring them at equilibrium at the distance where the potential is minimal and there is no force.

What about breaking it up into two (or more) forces and showing me each?
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XonatronNov 22 '13 at 16:28

You can split it up in many ways. In case of the Lennard-Jones potential $V(r)=Cr^{−12}−C′r^{−6}$, you can split it into the repulsive (-12) soft core part and the attractive (-6) van der Waals part. They do not have minima with positive $r$, hence do not lead to an equilibrium position. Only the combination balances repulsion and attraction at some equilibrium distance.
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Arnold NeumaierNov 23 '13 at 9:15

The strong force is "confined", it only extends a very small specific distance.

The weak force is "absorbed", it gets weaker faster than the inverse square law.

In summary, neither the strong force (involved in holding the protons together to make helium), or the weak force (involved in the decay of a neutron into a proton and electron) act at a distance subject to the inverse square law.

Your diagram of 2 H -> He is extremely misleading. First, He has Neutrons and can't be made from just 2 H. It need deuterium and tritium and it's a multi-step process involving first making deuterium via $\beta^{+}$ decay.

Also, your question and diagram seem to imply that there is a formula describing the curve you show but there isn't. Your curve is the aggregate of several different forces including electromagnetism (electrostatic repulsion) and the strong force (residual color force).

If you want the analogous "inverse square law" for nuclear forces, no such law exists. The electrostatic portion does behave a bit like an inverse square law but at short distances the Pauli exclusion principle and quantum mechanics dominate and make things complicated.

Matt Strassler has a great article (7 parts so far) on QFT and the strong force. The short version of it is that because quarks are so light we can't directly model the strong force. All of our predictions are educated guesses and we don't have any formulas to govern their macroscopic behavior. The other issue is that gluons interact with each other dashing any hope of an inverse-square-law-like formula.

Exactly, it's more than one formula, and what I want is all of those formulas to result in some sort of curve like this. Thanks for breaking it down for me. I guess what I am looking for does not exist.
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XonatronNov 22 '13 at 18:13